Knots and links $L\subset S^3$ carry a wealth of data. Spanning surfaces $F$ (1- or 2-sided), $\partial F=L$, especially {\bf checkerboard} surfaces from link diagrams $D\subset S^2$, help to mine this data. This text explores the structure of these surfaces, with a focus on a gluing operation called {\bf plumbing}, or {\it Murasugi sum}. First, naive classification questions provide natural and accessible motivation for the geometric and algebraic notions of essentiality (incompressibility with $\partial$-incompressibility and $\pi_1$-injectivity, respectively). This opening narrative also scaffolds a system of hyperlinks to the usual background information, which lies out of the way in appendices and glossaries. We then extend both notions of essentiality to define geometric and algebraic {\it degrees} of essentiality, $\underset{\hookrightarrow}{\text{ess}}(F)$ and $\text{ess}(F)$. For the latter, cutting $S^3$ along $F$ and letting $\mathcal{X}$ denote the set of compressing disks for $\partial (S^3\backslash\backslash F)$ in $S^3\backslash\backslash F$, $\text{ess}(F):=\min_{X\in\mathcal{X}}|\partial X\cap L|$. Extending results of Gabai and Ozawa, we prove that plumbing respects degrees of algebraic essentiality, $\text{ess}(F_1*F_2)\geq\min_{i=1,2}\text{ess}(F_i)$, provided $F_1,F_2$ are essential. We also show by example that plumbing does not respect the condition of geometric essentiality. We ask which surfaces de-plumb uniquely. We show that, in general, essentiality is necessary but insufficient, and we give various sufficient conditions. We consider Ozawa's notion of representativity $r(F,L)$, which is defined similarly to $\text{ess}(F)$, except that $F$ is a closed surface in $S^3$ that contains $L$, rather than a surface whose boundary equals $L$. We use Menasco's crossing bubbles to describe a sort of thin position for such a closed surface, relative to a given link diagram, and we prove in the case of alternating links that $r(F,L)\leq2$. (The contents of Chapter 4, under the title Alternating links have representativity 2, are first published in Algebraic \& Geometric Topology in 2018, published by Mathematical Sciences Publishers.) We then adapt these arguments to the context of spanning surfaces, obtaining a simpler proof of a useful crossing band lemma, as well as a foundation for future attempts to better classify the spanning surfaces for a given alternating link. We adapt the operation of plumbing to the context of Khovanov homology. We prove that every homogeneously adequate Kauffman state has enhancements $X^\pm$ in distinct $j$-gradings whose traces (which we define) represent nonzero Khovanov homology classes over $\mathbb{Z}/2\mathbb{Z}$, and that this is also true over $\mathbb{Z}$ when all $A$-blocks' state surfaces are two-sided. A direct proof constructs $X^\pm$ explicitly. An alternate proof, reflecting the theorem's geometric motivation, applies our adapted plumbing operation. Finally, we describe an interpretation of Khovanov homology in terms of decorated cell decompositions of abstract, nonorientable surfaces, featuring properly embedded (1+1)-dimensional nonorientable cobordisms in (2+1)-dimensional nonorientable cobordisms. This formulation contains a planarity condition; removing this condition leads to Khovanov homology for virtual link diagrams.

We consider integrable boundary states in the Heisenberg model. We begin by reviewing the algebraic Bethe Ansatz as well as integrable boundary states in spin chains. Then a new class of integrable states that was introduced last year by Caetano and Komatsu is described and expanded. We call these states the crosscap states. In these states each spin is entangled with its antipodal spin. We present a novel proof of the integrability of both a crosscap state that is known in the literature and one that is not previously known. We then use the machinery of the algebraic Bethe Ansatz to derive the overlaps between the crosscap states and off-shell Bethe states in terms of scalar products and other known overlaps. / Vi undersöker integrable gränstillstånd i Heisenbergmodellen. Vi börjar med att gå igenom den algebraiska Betheansatsen och integrabla gränstillstånd i spinnkedjor. Sedan beskrivs och expanderas en ny klass av integrabla tillstånd som introducerades förra året av Caetano och Komatsu. Vi kallar dessa tillstånd crosscap-tillstånd. I dessa tillstånd är varje spinn intrasslat med sin antipodala motsvarighet. Vidare presenterar vi ett nytt bevis av integrerbarheten hos både ett tidigare känt och ett nytt crosscap-tillstånd. Sedan använder vi den algebraiska Betheansatsens maskineri för att härleda överlappen mellan crosscap-tillstånden och off-shell Bethe tillstånd i termer av skalärprodukter och andra kända överlapp.